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500                                    ON  THE REFLECTION OF  LIGHT                                   [422
and similarly the second factor may be written with change of sign of rj
- {1 + AA'-t- T? (A + A')} {1 - AA'- ,7 (A'- A)}. Accordingly
.Dll, a ._ 1(1 + * A? - ^2 ( A + A')*] {(1 - AAQ2 - ^ (A- AQ*} smn a                                4^ A'2 (A2- I)2                             ' "A    '
In this, on restoring the values of A, A',
1 + A A'  r, (A + A') = 2^<s+6') {cos k(S + 8')rj cos k(S- 8')}, and
1 - A A' + 7? (A - A') = - 2^<W) {sin p (8 + S') + ?? sin /<; (S - 8%
Also                           4 A'2 (A2 - 1)2 = - 16ett(S+s/) sin2
and thus
 M   _ lco sin n." tt 
    , 7 r> ?72 sin2 ^ KG
x {sin2 p (5 + 8'} - tf sin2 i/c (S - 8')} ....... (55)
The transition between the two cases (of opposite behaviour when m = oo) occurs when sinh a = 0.    In general, this requires either
8')                       siniHS + 8')
'.           r\r          r> = -\ -- -  -  ' - -
'                      
conditions which are symmetrical with respect to 8 and 8', as clearly they ought to be*.    In (55), (56), if is limited to values less than unity.
Reverting to (43), we see that the evanescence of sinh2  requires that r   1 + t, or, if we separate the real and imaginary parts of r and t, r =  1 T <i + i<2.
If, for example, we take r =  I  t, we have
|r|2 = (l + 02 + ^=H-|i2+2^. Also                                         \r 2 = l-|i|2;
so that                              r|2 = l + ^,         t2 = -t1.
In like manner by interchange of r and t,
|/|2  l-J-r                     /y.|2_ _ ,
\ 1 1  j. -t- T-L ,             r |     r1}
showing that in this case r1} ^, are both negative.
The general equation (55) shows that sinh2 a is negative, when rf- lies between
sin2
This is the case (i) above denned where an increase in m leads to complete reflection.    On the other hand, sinh2 a is positive when if lies outside the
* That is -with reversal of the sign of 17, which makes uo difference here.unction of ?j. The first factor may be written